i.e., there exist some colorings \(s^{(0)},s^{(1)}\)  everywhere different that witness the pattern \(g(x_1,...,x_\ell ,s^{(0)}, s^{(1)})\)  with colors \((y_1,...,y_\ell )\) . The following lemma establishes this result. It follows the approach in [1]}.
